Theory and Applications of Categories, Vol. 25, No. 19, 2011, pp. 508–532.
COUNTABLE MEETS IN COHERENT SPACES
WITH APPLICATIONS TO THE CYCLIC SPECTRUM
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
Abstract. This paper reviews the basic properties of coherent spaces, characterizes
them, and proves a theorem about countable meets of open sets. A number of examples
of coherent spaces are given, including the set of all congruences (equipped with the
Zariski topology) of a model of a theory based on a set of partial operations. We also
give two alternate proofs of the main theorem, one using a theorem of Isbell’s and a
second using an unpublished theorem of Makkai’s. Finally, we apply these results to the
Boolean cyclic spectrum and give some relevant examples.
1. Introduction
A frame is a complete lattice in which finite infs distribute over arbitrary sups. We denote
the empty inf by ⊤ and the empty sup by ⊥, which are the top and bottom elements,
respectively, of the lattice. A map of frames preserves finite infs and arbitrary sups. The
motivating example of a frame is the open set lattice of a topological space. Moreover,
continuous maps induce frame homomorphisms. The result is a contravariant functor O
from the category Top of topological spaces to the category Frm of frames. A closed
subset D of a topological space is called indecomposable if it is not possible to write it
as a union of two proper closed subsets. A space is called sober if every indecomposable
closed set is the closure of a unique point, called the generic point of the set. On sober
spaces O is full and faithful.
If we let Loc denote the category of locales, which is simply Frm op , the opposite of
/ Loc .
the category of frames, this results in a covariant functor Top
In Section 2, we review basic properties of coherent spaces and prove a characterization
theorem which is similar to known results. Section 3 shows several ways in which coherent
spaces arise. A notable example concerns models of a first order theory described by
operations and partial operations. We show, for example, that the set of subobjects as
well as the set of congruences of a model, when equipped with a certain topology, called
the Zariski topology, give coherent spaces. In Section 4 we state and prove the main
theorem that shows that if X is a coherent space and {Ui } a countable family of open
The first author would like to thank NSERC of Canada for its support of this research. We would
all like to thank McGill and Concordia Universities for partial support of the second author’s visits to
Montreal.
Received by the editors 2011-06-14 and, in revised form, 2011-10-16.
Transmitted by Susan Niefeld. Published on 2011-11-19.
2000 Mathematics Subject Classification: 06D22, 18B99, 37B99.
Key words and phrases: countable localic meets of subspaces, Boolean flows, cyclic spectrum.
c Michael Barr, John F. Kennison, and R. Raphael, 2011. Permission to copy for private use granted.
⃝
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∩
subsets ∧
of X then the intersection Ui in the lattice of subspaces of X coincides with
the inf Ui in the lattice of sublocales of X. Section 5 discusses the connection with an
unpublished theorem of Makkai’s. In Section 6, we apply our results to the Boolean cyclic
spectrum and thus extend the work in [Kennison, 2002, Kennison, 2006, Kennison, 2009].
Section 7 gives examples.
1.1. Remark. In dealing with locales, it is standard to use “sublocale” to mean regular subobject. This means that sublocales correspond to regular quotients of frames.
Since Frm is equational, there is a one-one correspondence between regular quotients and
equivalence relations that are also models of the theory. Such equivalence relations are
called congruences. Thus if F is a frame there is a one-one correspondence between
congruences on F and sublocales of the locale L corresponding to F .
When we spoke on Theorem 4.1 of this paper at Category Theory 2011 in Vancouver,
André Joyal conjectured that this result was closely related to the property of being a
Baire space. We already knew that a space that satisfied Theorem 4.1 was Baire and
had an example of a Baire space that didn’t satisfy the theorem. So it was no surprise
that in the process of searching the literature, we discovered a paper by Till Plewe that
showed that the localic inf of a sequence of open sets in a space is spatial if and only if
every closed subset of X is Baire [Plewe (1996), Theorem 2.3]. In addition, we discovered
a theorem of John Isbell’s that shows that a locally compact (he called them locally
quasicompact) locale and intersections of descending sequences of such sublocales are
spatial [Isbell (1975), 4.1]. This result comes very close to proving our Theorem 4.1 as we
will see in the discussion following our proof. We feel that Theorem 4.1 and its related
results deserve to be better known because of their potential application to spectra, such
as the cyclic spectrum of a Boolean flow. We also mention some extensions of the theorem,
answering another question raised by Joyal, see 4.8.
2. Basic definitions and preliminary results
The results in this section are all standard and can be found, with somewhat different
proofs, in [Johnstone, 1982].
2.1. congruences and nuclei. A nucleus j on a frame F is a function j : F
such that
/F
Nuc-1. j is expansive: u ≤ j(u) for all u ∈ F ;
Nuc-2. j preserves finite inf;
Nuc-3. j is idempotent.
2.2. Theorem. There is a one-one correspondence between nuclei and congruences on a
frame.
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
Proof. Let F be a frame and j be a nucleus on F . Define a relation E by u E v if
j(u) = j(v). Since this clearly defines an equivalence relation, it is sufficient to show
it is closed under the frame operations. If u1 E v1 and u2 E v2 , it follows immediately
from Nuc-2, that (u1 ∧ u2 ) E (v1 ∧ v2 ). One might expect that showing that E is closed
under arbitrary sup would require that j preserve arbitrary sup, which it does not do in
general. From Nuc-2, it is clear that j is order preserving. Suppose we have ∨
two families
{u
{vα } such that ∨
uα E vα for ∨
all α. Then
∨ α } and ∨
∨ uα ≤ j(uα ) = j(vα ) ≤ j( vα ). Thus
uα ≤ j( vα ) so that j( uα ) ≤ j 2 ( vα ) = j( vα ) and the opposite inequality follows
by symmetry. Thus E is a congruence.
∨
Now suppose that E is a congruence on F . Define j by j(u) = {v | u E v}. It is clear
that j is expansive. Since E is closed under arbitrary sup, it is also clear that u E j(u)
from which it follows that j(u) E j 2 (u) so that j 2 (u) ≤ j(u). To see that j preserves finite
inf, we first show it preserves the partial order. If u1 ≤ u2 , then for all v with v E u1 , we
have (v ∨ u2 ) E (u1 ∨ u2 ) = u2 so that v ≤ v ∨ u2 ≤ j(u2 ). Since this is true of all such v, it
is true of their union which is j(u1 ). It is immediate that j(u1 ∧ u2 ) ≤ j(u1 ) ∧ j(u2 ). Next,
(u1 ∧ u2 ) E (j(u1 ) ∧ j(u2 )) and, by definition of j, it follows that j(u1 ∧ u2 ) ≥ j(u1 ) ∧ j(u2 ).
It is easy to verify that these processes are inverse to each other.
2.3. Coherent spaces. A topological space is said to be coherent if it is compact,
sober, the compact open sets are a base for the topology, and the intersection of two
compact open sets is compact.
If X is a topological space and M is a subbase for the topology on X, let N be the set
of complements of sets in M . We call the topology generated by M ∪ N the s-topology
(for strong topology). We will say that a subset of X is s-open, s-closed, or scompact, respectively, if it is open, closed, or compact, respectively, in the s-topology. It
is clear that open and closed sets are s-open and s-closed, respectively, while an s-compact
set is compact.
This s-topology as defined is an example of a topology gotten by beginning with a
subbase and adjoining the complements of the elements to get a larger subbase. Such
a topology is called a patch topology and we will adopt this terminology. We will
sometimes call the original topology the w-topology (for weak). We note that this
construction of a patch topology depends on the given subbase and is thus not a topological
construction. For example, on a Stone space, you recover the original topology if you take
for a subbase all the clopens, but if you take all opens, you just get the discrete topology.
2.4. Theorem. A topological space X is coherent if and only if it has a subbase M
consisting of compact sets and such that the topology generated by M and the complements
of the sets in M is compact Hausdorff.
Proof. The forward implication is based on the proof of [Hochster, 1969, Theorem 1]. Let
M denote the set of all compact open sets. The definition of coherent implies that M is
closed under finite meets and it is obviously closed under finite joins. This implies that N
is closed under finite meet and join as well. Since M ∪ N is closed under complementation
it is also a subbase for the closed set lattice in the s-topology, By dualizing [Kelley, 1955,
COUNTABLE MEETS IN COHERENT SPACESWITH APPLICATIONS TO THE CYCLIC
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Theorem 4.6], it will suffice to show that for any M0 ⊆ M and any N0 ⊆ N , if M0 ∪ N0
has the finite intersection property (FIP), then it has a non-empty meet. We will do this
using a series of claims.
We can assume that M0 is closed under finite meets and that N0 is maximal. The first is
trivial, while the second follows readily from the fact that the join of any chain of families
with the FIP has that property, since the FIP is determined by the finite subfamilies.
∩
The meet D = N ∈N0 N ̸= ∅ and meets every M ∈ M0 so that the family M0 ∪ {D} has
the FIP. Fix M ∈ M0 . The family {M ∩ N | N ∈ N0 } certainly has the FIP and is a
family of s-closed subsets of the compact space M .
D is indecomposable. Suppose D = D1 ∪ D2 with D1 and D2 closed subsets of D. At
least one of M0 ∪ {D1 } and M0 ∪ {D2 } has the FIP. Suppose that M0 ∪ {D1 } has the
FIP. Since D1 is closed, it is an intersection of sets in N . These sets can be added to N0
without destroying the FIP ∩
in M0 ∪ N0 and, by maximality, must already belong to N0 .
But this implies that D1 ⊇ N ∈N0 N and hence D = D1 .
The generic point x of D is in every M ∈ M0 . For if x ∈
/ M ∈ M0 , then D − M would
be a proper closed subset of D that contained x.
This completes the proof of the forward implication. For the converse, we begin by noting
that the sets in M are s-closed and hence s-compact and the same is true of their finite
meets since, by hypothesis, the s-topology is Hausforff. So we may suppose that M is
closed under finite meets and joins. By assumption, M is a base for the topology and we
easily see that every compact open set belongs to M .
Since X is s-compact, it is also compact. In view of the above definition of coherent
spaces, it suffices to show that X is sober. It is clear that X is T0 for if x, y ∈ X are such
that for every M ∈ M we have x ∈ M if and only if y ∈ M , then the same is true for the
family consisting of all sets in M ∪ N . Since this family forms a base for the s-topology,
which is Hausdorff, it follows that x = y. We denote by x̄, the closure of {x}. Let A be a
closed, indecomposable subset. We have to find a point p, necessarily in A, such that A
is p. Since X is T0 , such a point is unique if it exists. Assume that no such point p exists.
Then for every a ∈ A, we can choose a point φ(a) ∈ A such that φ(a) ∈
/ a. Since φ(a) ∈
/ a,
there exists a basic neighbourhood, Ma ∈ M , of φ(a) which misses a. Then a is in ¬Ma ,
the complement of Ma . Since ¬Ma is s-open and A is s-closed, hence s-compact, there is a
finite subset F ⊆ A such that A is covered by {¬Ma | a ∈ F }. Assume that F is chosen as
small as possible. The set F cannot consist of a single element
/∪
¬Ma . But if
∪ since φ(a) ∈
F = F1 ∪F2 is the union of two non-empty subsets, then A ⊆ ( a∈F1 ¬Ma )∪( a∈F2 ¬Ma ).
But these are closed sets and A is indecomposable, so it must be contained in the one or
the other set. This contradicts the assumption that F was chosen as small as possible.
2.5. Proposition. Suppose X is a coherent space with base M of compact∩open sets.
Suppose {Mα } is a family of sets from M and U is an open subset
∩ of X. If Mα ⊆ U ,
then for some finite set, say α1 , . . . , αm of indices, we have that m
i=1 Mαi ⊆ U .
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
Proof. The sets Mα are s-closed in a compact
space. The
∩
∩ set U is open, hence s-open
and therefore each Mα − U is closed. If Mα ⊆ U , then (Mα − U ) = ∅, whence a finite
intersection of them is empty.
3. Examples of Coherent Spaces
This section shows that coherent spaces arise in many ways. Often the proof that a given
space is coherent is omitted because it easily follows from the definition or from Theorem
2.4.
3.1. Notation. Whenever X is a given coherent space, M will denote the base of all
compact open subsets and N will denote the family of all sets whose complements are in
M . When constructing a coherent space, M will denote a family satisfying the conditions
of Theorem 2.4 and, after closing M up under finite joins and meets, N will denote the
family of all sets whose complements are in M .
3.2. Example. Any s-closed subspace of a coherent space is coherent.
3.3. Example. Let X be coherent and let M and N be as above. Then X with the
topology generated by N is coherent. We call the topology generated by N the dual of
the original topology, generated by M .
3.4. Definition. Let S be any set and let 2S be the family of all subsets of S. For each
a ∈ S let M (a) = {A ⊆ S | a ∈ A} and M = {M (a) | a ∈ S}. Then:
1. the w-topology on 2S is the one generated by the subbase M ;
2. the s-topology on 2S is generated by M together with N , the family of all complements of members of M ;
3. if F ⊆ 2S then the w-topology (respectively the s-topology) on F is the relative
topology on F obtained from the w-topology (resp. the s-topology) on 2S .
3.5. Example. The space 2S , with the w-topology, is coherent.
In general, if F ⊆ 2S is an s-closed subset, then F , with the w-topology, is also
coherent.
Proof. The w-topology on 2S is generated by M as defined above. In view of Theorem
2.5, it suffices to observe that the s-topology, generated by M ∪ N is the product topology,
obtained by regarding 2S as a product of S copies of 2 (where 2 is the discrete space with
two points). The assertion about F follows from 3.2.
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3.6. Notation. If U is an ultrafilter on 2S , then AU ⊆ S denotes the subspace for which
a ∈ AU if and only if M (a) ∈ U .
3.7. Example. Let F ⊆ 2S be given and suppose that AU ∈ F whenever U is an
ultrafilter on 2S with F ∈ U . Then F , with the w-topology, is coherent.
Proof. It suffices to observe that AU is the limit of U in the s-topology on 2S . So the
given condition implies that F is an s-closed subset of 2S .
3.8. Definition. Let T be a first order theory. We will say it is generated by finitary
partial operations if there is a family Ω = {Ω0 , Ω1 , . . . , Ωn , . . .} of sets such that an algebra
/ S for each n ∈ N and each ω ∈ Ωn .
S for T is given by a partial function ωS : S n
These partial operations may be subject to equations and Horn clauses, but they play no
role in the construction.
3.9. Example. Let S be a T-algebra. Let F be the family of all subsets of F ⊆ S which
satisfy (finitary) first-order conditions built up from equality, the operations of T , the
conditions x ∈ F and closed under binary infs, sups and negation. Then F , with the
w-topology, is coherent.
Such families would include T-subalgebras and, T-congruences, in case S = R × R
where R is a model of a T-algebra.
Proof. Such a family F satisfies the condition that whenever U is an ultrafilter on 2S
and F ∈ U , then AU ∈ F .
3.10. Example. If R is a ring, then the set P of all prime ideals of R, with the w-toplogy,
is coherent. The dual of this space is the set P with the Zariski topology.
3.11. Remark. It is shown in [Hochster, 1969] that every coherent space X is homeomorphic to a space of the form P (as in the above example, with the w-topology) and
also homeomorphic to a space P with the Zariski topology.
4. Countable meets
Recall from 1.1 that a sublocale is a regular subobject in the category of locales and that
there is a one-one correspondence between nuclei and congruences on a frame. If the
frame is O(X), the lattice of open
∪ sets of a space X, and if A ⊆ X is a subspace, then the
nucleus jA defined by jA (U ) = {V ∈ O(X) | V ∩ A ⊆ U } corresponds to the subframe
O(A).
of
4.1. Theorem. Suppose that X is coherent
∩ and suppose {Ui } is a countable sequence
∧
open sets. Then their spatial intersection Ui coincides with their localic inf Ui .
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
∩
Proof.
Let
M
be
the
base
of
compact
open
sets.
Suppose
A
=
Un and that Un =
∪
∧
Un and denote∨by jn , JA , and jL , resp.
σ∈Σn Mn,σ with each Mn,σ ∈ M . Let L =
the nuclei corresponding to Un , A, and L. By definition, jL = jn , the sup taken in the
lattice of nuclei. Since A ⊆ Un for all n, we see that jn ∪
≤ jA whence jL ≤ jA .
/
By a choice function, we mean a map ξ ∩
:N
Σn such that ξ(n) ∈ Σn for all
n > 0. If ξ is a choice function, we let Aξ = ∞
M
n,ξ(n) . Then from Mn,ξ(n) ⊆ Un , it
n=1
follows that Aξ ⊆ A.
If we suppose that A
L, (in the lattice of sublocales of X) then jL
jA . Thus
there is an open set V such that jL (V ) $ jA (V ) and hence there is an M0 ∈ M with
M0 ⊆ jA (V ) while M0 ̸⊆ jL (V ). This last implies that for all n > 0, M0 ̸⊆ jn (V ) which,
we will show, leads to a contradiction.
4.2. Lemma. Suppose that M ∈ M with M ̸⊆ jL (V ). Then for each n > 0, there is a
σ ∈ Σn such that M ∩ Mn,σ ̸⊆ jL (V ).
∨
Since M ̸⊆ jL (V ) = jL2 (V
)
and
j
=
jn , we see that M ̸⊆ jn (jL (V )) and hence
L
∪
M ∩ Un ̸⊆ jL (V ). But Un = σ∈Σn Mn,σ so there must be some σ ∈ Σn with M ∩ Mn,σ ̸⊆
jL (V ).
We will use this lemma to construct a choice function ξ such that M0 ∩∪Aξ ̸⊆ jL (U ).
Assuming this can be done, it follows from Aξ ⊆ A that M0 ∩ A = M0 ∩ ξ∈Ξ Aξ ̸⊆ U
from which we conclude that M0 ̸⊆ jA (U ) in contradiction to our supposition.
In this proof we use the standard notation := to mean “defined as”.
By the lemma, there is a ξ(1) ∈ Σ1 such that M1 := M0 ∩ M1,ξ(1) ̸⊆ jL (U ). Since M
is closed under finite meets, it follows that M1 ∈ M . Another application of the lemma
allows us to find a ξ(2) ∈ Σ2 such that M2 := M1 ∩ M2,ξ(2) ̸⊆ jL (U ). Since no term in the
descending chain
M0 ⊇ M1 ⊇ · · · ⊇ Mn ⊇ · · ·
∩
is included in jL (U ), it follows from Proposition
2.5
that
Mn ̸⊆ jL (U ). Since Mn ⊆
∩
Mn,ξ(n) it also follows that M0 ∩ Aξ = M0 ∩ n∈N Mn,ξ(n) ̸⊆ U and hence that M0 ∩ A ̸⊆ U
which means that M0 ̸⊆ jA (U ), contrary to our assumption.
4.3. Connection with Isbell’s theorem. As mentioned in the introduction, Isbell
showed that a locally compact locale and an intersection of a descending sequence of
locally compact locales are spatial. This can be used to give a quick proof of Theorem
4.1. A coherent space is locally compact. Moreover, since the compact open sets form a
base for the topology, it is immediate that every open set is also locally compact. The only
thing to be settled is how to replace the sequence of open sets by a descending sequence.
4.4. Proposition. If A and B are subspaces of a space X, then the sup jA ∨ jB in the
lattice of nuclei on O(X) is the smallest nucleus containing the composite jA jB .
Proof. Clearly jA ∨ jB is the least nucleus containing the set theoretic union jA ∪ jB .
The conclusion follows immediately from jA ∪ jB ⊆ jA jB ⊆ (jA ∪ jB )2 .
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Since jA jB is expansive and preserves finite inf, it is clear that the nucleus it generates
is the transfinite iteration of the powers of jA jB . By the above inequality, it is also the
transfinite iteration of the powers of jA ∪ jB even though the result does not obviously
preserve finite inf.
4.5. Corollary. If A is open or B is closed, then jA∩B = jA ∨ jB .
Proof. We know that for open U and V we have V ⊆ jA∩B (U ) if and only if V ∩A∩B ⊆ U .
If A is open then (V ∩A)∩B ⊆ U if and only if V ∩A ⊆ jB (U ) if and only if V ⊆ jA jB (U ).
If B is closed, then V ∩ A ∩ B ⊆ U if and only if V ∩ A ⊆ ¬B ∪ U = jB (U ) if and only if
V ⊆ jA jB (U ). In either case we see that jA∩B = jA jB whence jA jB is already a nucleus
and therefore is jA ∨ jB .
We can now complete the second proof of Theorem 4.1. For if {Un } is a sequence of
open sets in X, we can replace them by U1 , U1 ∩ U2 , U1 ∩ U2 ∩ U3 , . . . to get a descending
sequence. As already noted, these will all be locally compact when X is coherent.
4.6. Extension to locally closed sets. Joyal also raised the question of extending
the result of Theorem 4.1 to an intersection of locally closed sets. We will see that this is
correct.
4.7. Proposition. An intersection of closed sets is spatial.
∩
Proof. Suppose that Fα is a family of closed sets and F = Fα . If j is any nucleus for
which j ∪
≥ jFα for all α, then for any α and any∨open U we have j(U ) ⊇ ¬Fα ∪ U so that
j(U ) ⊇ (¬Fα ) = ¬F ∪ U = jF (U ) so that jFα ≥ jF while the reverse inequality is
evident.
4.8. Corollary. In any space in which every localic inf of every sequence of opens is
spatial, the same is true of every localic inf of locally closed subsets.
Proof. Let
Un ∩ Fn with Un open and Fn closed.
∩ {Bn } be such
∩ a sequence.
∧ Write∧Bn = ∧
Let A = Un and F = Fn . Since Bn = Un ∧ Fn = A∧F = A∩F , the conclusion
follows.
5. Connections with a Theorem of Makkai’s
Theorem 4.1 can be derived from an unpublished theorem of Michael Makkai’s that extends a famous result of [Rasiowa & Sikorski, 1950], which can also be found in [Rasiowa
& Sikorski, 1968, p. 88]. In order to discuss the connection, we need to recall a few basic
/L
concepts. If L is a locale, then a point of L is defined to be a localic map p : 1
where 1 stands for the one-point topological space (regarded as a locale). In other words,
/ {⊥, ⊤}. For example, if L = O(X) is a
a point of L is a frame homomorphism p : L
spatial locale, then every element x ∈ X determines a point x
b for which x
b(U ) = ⊤ if and
only if x ∈ U .
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
5.1. Enough points We say that the locale L has enough points if whenever x ̸= y
/ {⊥, ⊤} for which p(u) ̸= p(v). Recall that a locale is spatial if
there is a point p : L
and only if it is isomorphic to the locale of all open subsets of a topological space. The
following straightforward result is well-known:
5.2. Proposition. A locale is spatial if and only if it has enough points.
Proof. If L is isomorphic to O(X) where X is a topological space, then it clearly has
enough points of the form x
b for x ∈ X.
Conversely, assume that L has enough points. Let pt(L) be the set of all points of L
and for each u ∈ L define an open subset u
b ⊆ pt(L) by u
b = {p ∈ X | p(u) = ⊤}. It
readily follows that L is isomorphic to O(pt(L)).
In the next proposition, we use the obvious fact that a T0 space is sober if and only
every indecomposable closed set has at least one generic point.
5.3. Proposition. The set theoretic meet of any family of sober subspaces of a T0 topological space is sober.
Proof. Let X be a T0 space and
∩ let {Yα } be a family of sober subspaces of X. Let
/
p:1
Y be any point of Y = Yα . Then, for each α, there is a corresponding point
/ Yα given by pα = iα p where iα is the inclusion Y ⊆ Yα . Since Yα is sober, the
pα : 1
point pα is represented by an element xα ∈ Yα . By factoring through the inclusion of
/ X we get a point of X which is represented by xα . Since X is T0 , the elements xα
Yα
must all coincide, and must therefore be in Y .
5.4. Proposition.
Let {Yα } be a family of sober subspaces of a T0 topological space X.
∩
Then, Y :=∧ Yα , the intersection of the family in the lattice of all subspaces coincides
with L :=∧ Yα , the intersection in the lattice of all sublocales of X, if and only if the
sublocale Yα has enough points.
Proof. If the two intersections coincide, then L is spatial so it must have enough points.
Conversely, assume that L has enough points. Since L is contained in each spatial sublocale Yα , we see that every point of L is a point of each Yα which, by sobriety, corresponds
to an element of Yα and hence to an element of Y . Let EYα , EY and EL denote the congruences on O(X) determined by Yα , Y , and L respectively. Since EL is the sup in the
lattice of congruences of the EYα and EYα ⊆ EY for all α, it is immediate that EL ⊆ EY .
/ / O(X)/ EY . But since the set P of points of L
Thus we have a surjection O(X)/ EL
coincides with the set |Y | we have that the bottom row of
O(X)/ EL
/ / O(X)/ EY
/ 2|Y |
2P
∼
=
is an isomorphism from which it is evident that the top map is also an isomorphism.
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There are lots of sober subspaces in view of the above proposition and:
5.5. Proposition.
1. Every closed subset of a sober space is sober.
2. Every open subset of a sober space is sober.
Proof.
1. Straightforward, by looking at closed indecomposable subsets.
2. Let X be a sober space and let U ⊆ X be an open subset. Let A0 =
̸ ∅ be a
(relatively) closed indecomposable subset of U . Let A be the closure of A0 in X.
We first claim that A is indecomposable in X. Suppose A = B ∪ C. Let B0 = B ∩ U
and C0 = C ∩ U . Then A0 = B0 ∪ C0 . Since A0 is indecomposable in U we see that
either A0 = B0 or A0 = C0 . Say A0 = B0 . Then the closure of B0 is contained in B
but the closure of B0 is the closure of A0 , which is A so A = B.
Now, since A is indecomposable in X there exists an element x ∈ X such that A is
the closure of {x}. It suffices to show that x ∈ U . Let u ∈ A0 . Then u ∈ A and
therefore is in the closure of {x}. Thus x is in every neighbourhood of u. Since U
is a neighbourhood of u we must have x ∈ U .
5.6. Proposition.
Let X be a topological space and let {Aα } be a family of closed subsets
∩
∧
of X. Then Aα , the intersection in the lattice of all subspaces of X, coincides with Aα ,
the intersection in the lattice of all sublocales of X.
Proof. For each α, let Wα ⊆ X be the complement of Aα . Then each Wα is obviously
open. Let jα be the nucleus of Aα , viewed as a sublocale of O(X). It is readily shown
that jα is given by jα (U ) = U ∨ Wα for all U ∈ O(X). Now let j be the
∨ sup of {jα } in
the lattice of all∩nuclei on O(X). It is readily shown that j(U ) = U ∨ ( Wα ) but this is
the nucleus for Aα , viewed as a sublocale of O(X).
5.7. Makkai’s Theorem. The theorem of Rasiowa and Sikorski mentioned at the beginning of this section can be paraphrased as follows.
5.8. Theorem. Let A be a Boolean algebra and Q be a countable family of subsets of A.
Let B be the Boolean algebra freely generated by A together with one element forced to be
a sup for each set in Q. Then there are enough 2-valued Boolean representations of B
that preserve all the sups from Q to separate the points of A.
Had the conclusion been that there were enough such “points” to separate the points
of B, this would have given a different proof of our Theorem 4.1 in the special case of
a Stone space. However, in a so-far unpublished work, Makkai has strengthened the
Rasiowa-Sikorski theorem in two ways: the theorem is generalized to meet semi-lattices
and the conclusion has been strengthened in the way required to give an alternate proof
of our theorem in the general case.
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
5.9. Theorem. [Makkai, unpublished] Assume that P is a meet-semi-lattice with a coverage system generated by Y1 ∪ Y2 where Y1 is a countable set of covers and Y2 is a set of
finite covers. Then the locale generated by these data (see [Johnstone, 1982, pp. 57–59])
has enough points.
We now sketch how this result can be used to give an alternate proof of 4.1.
Proof. We work∪in the meet-semilattice M of all compact open subsets of X. For each
i, we write Ui = Σi where Σi ⊆ M . We let Y1 be the countable set of covers given by
saying that Σi is a cover of the top element of O(X). We let Y2 be
∪ all covers of the form
(M, C) where M ∈ M and C ⊆ M is a finite subset for which C = M . It is readily
shown that the locale generated by the
∧ meet-semilattice M with the coverage∧system
generated by Y1 ∪ Y2 is the sublocale Ui . By Makkai’s result, it follows that Ui has
enough points, and the proof of this proposition then follows from 5.4.
6. Applications to the Boolean cyclic spectrum
In this section, we use the result about countable meets of open subsets of a coherent
space to obtain results about the cyclic spectrum of a Boolean flow. We briefly review the
earlier papers, [Kennison, 2002], [Kennison, 2006], explaining what a Boolean flow is, and
why we want to study it. We also review our notation (some of which has been changed
to make it more descriptive).
(Discrete) Dynamical Systems. Conceptually, imagine that we have a “system”
that can be in different states. Let X be the set of all possible states and assume that
X has a topology. Imagine further that, when in state x, the system will, after a fixed
/ X is conperiod of time, make a transition to a state t(x). We assume that t : X
tinuous. Then if the system starts out in state x its future states will form a sequence
{x, t(x), t2 (x), . . . , tn (x), . . .} called the orbit of x. Such an orbit might eventually cycle
or approach a cycle or behave chaotically and so forth. We want to break X down into
components which are “close” to being cyclic.
Formally, a flow in a category C is a pair (X, t) where X is an object of C and
/ X is a map of C. If (X, t) and (Y, s) are flows in C then a map h : X
/Y
t:X
is a flow homomorphism if ht = sh. This enables us to define Flow(C), the category
of flows in C. In view of the above comments, we wish to study the category of flows
on topological spaces. If (X, t) is such a flow, it is fruitful to assume that X is compact
and Hausdorff but further assuming that X is totally disconnected, and therefore a Stone
space, is unnecessary. However, the well-known method of symbolic dynamics can be
used to approximate a flow on any topological space by a flow on a Stone space. Roughly
speaking, given a flow on a space X, we write X as a finite union:
∪
X = {Aσ | σ ∈ Σ}
where each Aσ is a closed subset and Σ is a finite set of “symbols”. Then each x ∈ X
is “compatible” with at least one sequence (σ0 , σ1 , . . . , σn , . . .) meaning that tn (x) ∈ Aσn ,
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for all n ∈ N. (Since the subsets {Aσ } may overlap, each x may be compatible with more
than one such sequence of symbols. In practice, the family {Aσ } of subsets is chosen to
make the overlaps as small as possible.) The compatible sequences (i.e. sequences {σn }
for which there exists x ∈ X compatible with {σn }) form a subset of ΣN which is closed
/ ΣN where:
under the “truncation map” tr : ΣN
tr(σ0 , σ1 , σ2 , . . .) = (σ1 , σ2 , . . .)
(Note that if x is compatible with the sequence s then t(x) is compatible with the sequence
b is the closure of the set of all compatible sequences in ΣN then (X,
b tr) is a
tr(s).) If X
flow in Stone spaces which approximates the original flow (X, t). In what follows, we
will assume that (X, t) is a flow on a Stone space X. We will also, in some of our
b For
examples, denote the truncation map by t instead of tr and not use the notation X.
details, see [Kennison, 2006].
The Boolean Cyclic Spectrum. In view of the above, we will study flows on Stone
spaces. By the Stone Duality Theorem, it is obvious that the category of flows on Stone
spaces is dual to the category of flows on Boolean algebras. (Given as flow (X, t) on a
Stone space X, we let (B, τ ) denote the corresponding flow on the Boolean algebra B of
all clopen subsets of X where τ = t−1 ).
We define when a Boolean flow (B, τ ) is cyclic (see below) and this definition readily
extends to what is meant by a cyclic sheaf of Boolean flows over a locale. The cyclic
spectrum of (B, τ ) is then the “best” way to map B to the global sections of a cyclic
sheaf of Boolean flows over a locale (see below). This is analogous to the local ring sheaf
of a ring, see [Kennison, 2006] and [Johnstone, 2002] for details. We outline the main
construction, our notation and the main results of the previous papers
1. We let (B, τ ) denote a flow in Boolean algebras and let (X, t) be the corresponding
flow on Stone spaces, where B is the Boolean algebra of clopens of X.
2. We say that as Boolean flow (B, τ ) is cyclic if for all b ∈ B there exists n > 0 such
that τ n (b) = b.
b denote the profinite integers, which is the inverse limit of the groups
3. We let Z
/ Zn whenever m is a
{Zn | n ≥ 2} together with the obvious quotient maps Zm
multiple of n. The Boolean flow (B, τ ) is cyclic if and only if there is a compatible
b
/ B such that, for all n > 0 and all b ∈ B, we have α(n, b) = tn (b).
action α : Z×B
b
/X
This, in turn, is true if and only if there is a similar compatible action α : Z×X
where X is the Stone space corresponding to B.
4. We say that I ⊆ B is a “flow ideal” of (B τ ) if I is an ideal such that there is a
/ B. It readily follows
/ B/I which commutes with τ : B
homomorphism B/I
that a flow ideal is a Boolean ideal I ⊆ B satisfying the condition that whenever
b ∈ I then τ (b) ∈ I.
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
5. We let W be the space of all flow ideals of (B, τ ). For each b ∈ B, we define
N (b) ⊆ W by:
N (b) = {I ∈ W | b ∈ I}
We topologize W so that the family {N (b)} forms a base. We let O(W ) denote the
locale of all open subsets of W . By the argument given in Example 3.7, the space
W is coherent.
6. There is a canonical sheaf Q of Boolean flows over W where the stalk QI is defined
as B/I for each I ∈ W . (Note: In the previous papers, the sheaf Q was denoted by
B 0 . Here we use Q, which is, in a sense, the universal quotient flow of (B, τ ), see
[Kennison, 2006]).
7. We let O(W )cyc be the largest sublocale of O(W ) which forces Q to become cyclic.
That is, O(W )cyc is the largest sublocale such that, for each b ∈ B, the family
{N (b − τ k b)} covers O(W )cyc . The sheaf Q restricts to a sheaf Qcyc on the locale
O(W )cyc and Qcyc is a cyclic Boolean flow over the locale O(W )cyc . It is the “best”
such sheaf and is called the Boolean cyclic spectrum of (B, τ ). Previously,
O(W )cyc and Qcyc were denoted by Lcyc and B ∗ respectively.
8. A flow ideal I ∈ W is a cyclic flow ideal if the Boolean flow B/I is cyclic. We let
Wsp.cyc ⊆ W denote the subspace of all cyclic flow ideals. It is shown in [Kennison,
2006] that O(W )cyc is spatial (isomorphic to the locale of opens of a topological
space) if and only if it is isomorphic to O(Wsp.cyc ). But in [Kennison, 2006], Lcyc
is used for O(W )cyc and Wcyc is used for Wsp.cyc . To avoid confusion, the
notations Lcyc and Wcyc will not be used in this paper.
9. We let Γ(Qcyc ) denote the set of global sections over the cyclic spectrum.
10. For further details, consult the previous papers, particularly [Kennison, 2006].
b and proved
In the previous papers, we proved the results involving the actions by Z
that Q and Qcyc have the universal properties mentioned above. We described Q and Qcyc
in the case that (B, τ ) is finitely generated (as a flow). We also pursued the questions
of whether O(W )cyc is spatial and of how to describe the global sections over the cyclic
spectrum. In this paper, using Theorem 4.1, we will:
1. show that the cyclic spectrum of a countable flow is always spatial;
2. give an explicit description of the nucleus jcyc ;
3. show that the cyclic spectrum always has the Lindelöf property;
4. describe Γ(Qcyc ), the set of global sections over the cyclic spectrum.
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The cyclic spectrum of a countable flow is spatial. The following extends the
result in [Kennison, 2006] that the cyclic spectrum of a finitely generated Boolean flow
is spatial. Example 7.1 shows that for uncountable B, the cyclic spectrum need not be
spatial, which answers a question left open in [Kennison, 2006].
6.1. Proposition. If B is countable then the locale O(W )cyc is spatial (so we can regard
Qcyc as a sheaf over the space Wsp.cyc and we say that B has a spatial cyclic spectrum).
Proof. The base of the cyclic spectrum, O(W )cyc , can be defined as the largest sublocale
of O(W ) for which every b ∈ B becomes cyclic, meaning that, for each such b, the basic
open sets {N (b − τ k b)} cover O(W )cyc . Note that N (b) is defined
above; for more details,
∪
see [Kennison, 2006]. It follows that, if we let cyc(b) = {N (b − τ k b) | k > 0}, then
O(W )cyc is the localic meet of {cyc(b) | b ∈ B}. If B is countable, then this meet is
spatial by Theorem 4.1.
For technical reasons, we want to generalize the above result. We need the following
definition.
6.2. Definition. Let C ⊆ B be a countable subset. Let WC be the largest sublocale of
O(W ) which makes every c ∈ C cyclic. That is, WC is the localic meet of {cyc(c) | c ∈ C}.
Furthermore, we say that a flow ideal I ∈ W is C-cyclic if for every c ∈ C, there exists
k > 0 such that I ∈ N (c − τ k c).
6.3. Proposition. The sublocale WC is spatial for every countable C ⊆ B. The sublocale
WC ⊆ W can be identified with the subspace of all C-cyclic flow ideals.
Proof. The proof of the previous proposition clearly applies here.
6.4. Remark. We will routinely identify WC with the subspace of all C-cyclic flow ideals.
Description of the nucleus jcyc and the Lindelöf property. Our next result
gives a fairly technical, but quite useful, characterization of jcyc . We first need some
definitions and notation.
6.5. Definition. An open set U ∈ O(W ) is countably basic if we can write U as a
countable union of basic open subsets of the form N (b) for b ∈ B.
6.6. Theorem. Let (B, τ ) be a Boolean flow and let b ∈ B and U ∈ O(W ) be given.
Then N (b) ⊆ jcyc (U ) if and only if there exists a countable subset C ⊆ B and a countably
basic open set U0 ⊆ U such that N (b) ∩ WC ⊆ U0 .
Proof. We define J(U ) as the union of all N (b) for which there exists a countable subset
C ⊆ B and a countably basic open set U0 ⊆ U such that N (b) ∩ WC ⊆ U0 . We claim that
J is a nucleus. The only non-trivial step is proving that J is idempotent. By examining
the nucleus jC for the subspace WC ⊆ W , it readily follows that N (b) ∩ WC ⊆ U0 if and
only if N (b) ⊆ jC (U0 ). Assume N (b) ⊆ J(J(U )). Then there exists a countable
∪ subset
C ⊆ B and a countably basic set V ⊆ J(U ) such that N (b) ⊆ jC (V ). Write V = N (dn )
where N (dn ) ⊆ J(U ) for all n ∈ N. Then for each n, there exists a countably basic
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
∪
Vn ⊆ U and
∪ a countable subset C(n) ⊆ B with N (dn ) ⊆ jC(n) (Vn ). Let U0 = Vn and
D = C ∪ C(n). It suffices to show that N (b) ⊆ jD (U0 ). But jD ≥ jC and jD ≥ jC(n)
for all n. So V ⊆ jD (U0 ) and jC (V ) ⊆ jD (jD (U0 )) = jD (U0 ) and the ∪
claim follows.
The nucleus J makes every b ∈ B cyclic (let C = {b} then U0 = N (b − τ n b) covers
WC and is countably basic). It follows that J ≥ jcyc and the opposite inclusion, J ≤ jcyc
is obvious.
6.7. Definition.∨A locale L has the Lindelöf property if whenever F ⊆ L covers L
(that is whenever F = ⊤) then F has a countable subset F0 ⊆ F which also covers L.
6.8. Proposition. The locale O(W )cyc has the Lindelöf property.
Proof. It suffices to show that
∪any cover of O(W )cyc by basic opens N (b) has a countable
subcover. Suppose that U = N (bα ) and that U covers O(W )cyc . Then jcyc (U ) = ⊤ =
N(0) so,
∪ by the above theorem, there is a countably basic
∪ U0 ⊆ U with jcyc (U0 ) = ⊤. Let
U0 = N (cn ). Then for each n we have N (cn ) ⊆ N (bα ) which readily implies that
there exists α with N (cn ) ⊆ N (bα ) and so only a countable set of the N (bα ) is needed to
cover U0 and hence to cover O(W )cyc .
Description of the global sections over the cyclic spectrum. It remains to
describe the Boolean flow Γ(Qcyc ) of all global sections over the cyclic spectrum of B. We
first do this when B is countable then show how to extend that result to arbitrary B.
We will use the following convenient assumption for the rest of this section.
6.9. Assumption. Let (B, τ ) be a Boolean flow. By Stone duality, we can suppose that
B = clop(X), the algebra of clopen sets of the Stone space X. Another use of Stone
/ X such that τ (b) = t−1 (b)
duality shows that there is a unique continuous map t : X
for all b ∈ B. Note that we are not merely assuming that B is isomorphic to
clop(X) but that every b ∈ B is actually a clopen set of X . See [Kennison, 2002]
for more details.
6.10. Definition. Let X be as above, let b ∈ B be a clopen subset of X and let x ∈ X be
given. We say that x is k-cyclic with respect to b if, for all n ≥ 0, we have tn (x) ∈ b
if and only if tn+k (x) ∈ b.
We say that x is cyclic with respect to b if x is k-cyclic with respect to b for some
k > 0 (in this case, k is a period of x).
Further, x ∈ X is cyclic if, for all b ∈ B, x is cyclic with respect to b. We let Xcyc
denote the subspace of all cyclic elements of X.
We let k-Cy(b) denote the set of all x ∈ X which are k-cyclic with respect to b.
6.11. Examples. Suppose (B, τ ) is a cyclic Boolean flow, meaning that for every b ∈ B
there exists k > 0 such that b = τ k b. Then X = Xcyc .
Another example is given by (X, t) where X = {0, 1}N and t is the truncation map.
Then x ∈ Xcyc if and only if x is a periodic sequence. See Example 7.5 and the definition
that precedes it for details.
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6.12. Definition. Let I ⊆ B be a flow ideal. Then I corresponds to the flow quotient
B/I which, by Stone duality, corresponds to a closed subflow A(I) ⊆ X.
If b ∈ B, we let ⟨b⟩ denote the flow ideal generated by b. By abuse of language, we use
A(b) to denote A(⟨b⟩).
6.13. Lemma.
1. Let I ⊆ B be a flow ideal. Then A(I) =
b ∩ A(I) = ∅.
∩
{¬b | b ∈ I}. Also, b ∈ I if and only if
2. Let I, J ⊆ B be flow ideals. Then I ⊆ J if and only if A(J) ⊆ A(I).
3. Let b ∈ B be given and regard b as a clopen subset of X. Then x ∈ A(b) if and only
if tn (x) ∈
/ b for all n ≥ 0.
Proof.
1. First, we show that if A ⊆ X is a closed subflow, then the corresponding flow ideal is
/ X be the inclusion map. Then i−1 : clop(X)
/ clop(A)
{b | b∩A = ∅}. Let i : A
−1
is the corresponding quotient of B = clop(X). Obviously i (b) = 0 if and only if
A ∩ b = ∅.
It follows∩that if A is the closed subflow that corresponds
to the flow ideal I, then
∩
A(I) ⊆ {¬b | b ∈ I}. It is readily checked that {¬b | b ∈ I} is topologically
closed
∩ and closed under the action of t (as I is closed under the action of τ ). Suppose
d ∩ {¬b | b ∈ I} = ∅. We must show that d ∈ I. It follows that d is covered by the
elements of I and, by compactness, by a finite subset of I. Since I is closed under
finite unions, there exists b ∈ I with d ≤ b and this implies that d ∈ I.
2. Straightforward, in view of the first paragraph, above.
3. Clearly A(b) is the largest closed subflow of X which is disjoint from b. A straightforward check shows that the given description of A(b) has this property.
6.14. Lemma. Let b ∈ B and k > 0 be given. Then k-Cy(b) = A(b − τ k b).
Proof. Straightforward.
6.15. Lemma. Let b ∈ B and k > 0 be given. Then for every non-zero multiple m of k,
we have ⟨b − τ m b⟩ ⊆ ⟨b − τ k b⟩.
Proof. It clearly suffices to show that if I ⊆ B is a flow ideal and (b − τ k b) ∈ I then
(b − τ m b) ∈ I. But suppose (b − τ k b) ∈ I. By applying τ k , we see that (τ k b − τ 2k b) ∈ I.
Adding (b − τ k b) to it gives us (b − τ 2k b) ∈ I and the result follows by an easy induction.
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MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
6.16. Proposition. For I ∈ W , we have I ∈ Wsp.cyc if and only if A(I) ⊆ Xcyc .
Proof. Assume I ∈ Wsp.cyc and let x ∈ A(I) be given. To prove that x ∈ Xcyc , suppose
b ∈ B. Since I is a cyclic flow ideal, there exists k > 0 such that (b − τ k b) ∈ I. It readily
follows that A(I) ⊆ A(b − τ k b) and, in view of lemma 6.14, we see that x is k-cyclic with
respect to b. Since b is an arbitrary member of B, we see that x ∈ Xcyc .
Conversely, assume A(I) ⊆ Xcyc and that b ∈ B is given. It easily follows from Lemma
6.14 that {¬(b − τ k b) | k > 0} covers Xcyc . Since A(I) ⊆ Xcyc it is covered by a finite set
{¬(b − τ k(i) b)}. Let m be a common multiple of the set {k(i)}, then it is readily shown
that A(I) ⊆ ¬(b − τ m b) which shows that (b − τ m b) ∈ I.
6.17. Proposition. Assume that B is countable (or more generally, that B has a spatial
cyclic spectrum). Let d ∈ B be given. Let db denote the corresponding constant section in
Γ(Q) and let dbcyc denote the restriction of db to the subspace Wsp.cyc . Then dbcyc = 0 if and
only if d ∩ Xcyc = ∅.
Proof. Recall that d is a clopen subset of X. Assume that d ∩ Xcyc = ∅. Let I ∈ Wsp.cyc
be given. As shown above, A(I) ⊆ Xcyc so d ∩ A(I) = ∅ and therefore d ∈ I. Since d ∈ I
for all I ∈ Wsp.cyc , it follows that dbcyc , the restriction of db to Wsp.cyc is 0.
Conversely, assume that dbcyc = 0 and that x ∈ d ∩ Xcyc . We need to derive a contradiction. Since x ∈ Xcyc , we can, for every b ∈ B, find a positive integer k(b) such that x is
k(b)-cyclic with respect to b. This implies that tn (x) ∈
/ (b − τ k(b) b) for all n ≥ 0. Let I be
the set of all c ∈ B such that tn (x) ∈
/ c for all n ≥ 0. Then I is readily seen to be a flow
ideal of B and a cyclic flow ideal as (b − τ k(b) b) ∈ I for all b ∈ B. Moreover, x ∈ A(I) so
b ̸= 0 which contradicts
d ∩ A(I) ̸= ∅ as x ∈ d ∩ A(I). So d ∈
/ I and this implies that d(I)
the assumption that dbcyc = 0.
6.18. Proposition. Let c, d ∈ B be given (and regard each element of B as a clopen
subset of X). Then
N (c) ∩ Wsp.cyc ⊆ N (d) if and only if A(c) ∩ Xcyc ⊆ A(d)
Proof. First, assume A(c) ∩ Xcyc ⊆ A(d). Let I ∈ N (c) ∩ Wsp.cyc be given. We need to
show that d ∈ I. Since c ∈ I, we have ⟨c⟩ ⊆ I so A(I) ⊆ A(c). By Proposition 6.16, we
have A(I) ⊆ Xcyc , so, by our assumption, A(I) ⊆ A(d). But then, by Lemma 6.13, (2),
⟨d⟩ ⊆ I and d ∈ I.
Conversely, assume N (c) ∩ Wsp.cyc ⊆ N (d). Let x ∈ A(c) ∩ Xcyc be given. Since
x ∈ Xcyc , we can choose, for each b ∈ B, an integer k(b) > 0 such that x is k(b)-cyclic
with respect to b. Let I be the flow ideal generated by c and {b−τ k(b) b | b ∈ B}. Then I is
the smallest flow ideal containing c and each b −τ k(b) b so A(I) is the largest
∩closed subflow
contained in A(c) and each A(b − τ k(b)∩
b) which means that A(I) = A(c) ∩ b A(b − τ k(b) b).
By the choice of k(b), we have x ∈ b A(b − τ k(b) b) and we assumed that x ∈ A(c) so
x ∈ A(I). Clearly tn (x) ∈ A(I) for all n ≥ 0. But by our assumption that N (c)∩Wsp.cyc ⊆
N (d), we see that d ∈ I so d ∩ A(I) = ∅ so tn (x) ∈
/ d (as tn (x) ∈ A(I)) which implies that
x ∈ A(d).
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6.19. Corollary. Let c1 , c2 , d ∈ B be given. Then:
N (c1 ) ∩ N (c2 ) ∩ Wsp.cyc ⊆ N (d) if and only if A(c1 ) ∩ A(c2 ) ∩ Xcyc ⊆ A(d)
Proof. This follows from the above proposition with c = c1 ∨ c2 . Note that N (c1 ∨ c2 ) =
N (c1 ) ∩ N (c2 ) and A(c1 ∨ c2 ) = A(c1 ) ∩ A(c2 ).
6.20. Definition. A subset S ⊆ Xcyc is rectified by b ∈ B if there exists d ∈ B such
that
S ∩ A(b) = d ∩ A(b) ∩ Xcyc
We let Rect(S) denote the set of all b ∈ B which rectify S. We say that S ⊆ Xcyc is
regular if
∪
Wsp.cyc ⊆ {N (b) | b ∈ Rect(S)}.
6.21. Proposition.
1. If d ∈ B, then d ∩ Xcyc is a regular subset of Xcyc .
2. The regular subsets of Xcyc are closed under complementation (within Xcyc ) and
under finite unions and intersections (which includes the empty subset and Xcyc
itself ).
Proof.
1. First, it is clear that every b ∈ B rectifies d ∩ Xcyc and when b = ⊥, then N (b)
covers W .
2. Closure under complementation follows by verifying that b rectifies S if and only
if b rectifies Xcyc − S. To prove closure under pairwise intersections, it suffices to
verify that if b ∈ Rect(S) and c ∈ Rect(T ), then b ∨ c ∈ Rect(S ∩ T ). Note that if S
is empty, then every b ∈ B rectifies S. The rest of the proof follows by considering
complements within Xcyc .
6.22. Notation. Assume that B is countable (or more generally, that B has a spatial
cyclic spectrum). We let Reg(B) denote the Boolean algebra of all regular subsets of Xcyc .
In view of (1) of the above proposition, there is a canonical Boolean homomorphism from
B to Reg(B).
6.23. Lemma. If d, e ∈ B are given, then d ∩ A(d − e) ⊆ e.
/ e. Then
Proof. Assume the contrary, that there exists x ∈ d ∩ A(d − e) but with x ∈
x ∈ (d − e) ∩ A(d − e) which is a contradiction.
526
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
6.24. Theorem. Assume that B is countable (or more generally, that B has a spatial cyclic spectrum). The Boolean algebra Γ(Qcyc ) is canonically isomorphic to Reg(B).
/ Reg(B) mentioned above and
Moreover, the isomorphism commutes with the map B
b
/ Γ(Qcyc ) which sends d ∈ B to d.
the map B
Proof. Let σ ∈ Γ(Qcyc ) be given. Locally, σ agrees with constant sections of the form
db so we can find a family {(dα , bα )} such that σ = dbα on N (bα ) ∩ Wsp.cyc . It follows that
these sections are compatible, meaning that
N (bα ) ∩ N (bβ ) ∩ Wsp.cyc ⊆ N (dα − dβ )
∪
Now define S ⊆ Xcyc as α (dα ∩ A(bα ) ∩ Xcyc ). We claim that each bβ rectifies S. We
must show that S ∩ A(bβ ) = dβ ∩ A(bβ ) ∩ Xcyc . We have:
∪
S ∩ A(bβ ) = (A(bβ ) ∩ dα ∩ A(bα ) ∩ Xcyc )
α
We note that if β = α then (A(bβ ) ∩ dα ∩ A(bα ) ∩ Xcyc ) reduces to dβ ∩ A(bβ ) ∩ Xcyc ,
so it suffices to show in general that (A(bβ ) ∩ dα ∩ A(bα ) ∩ Xcyc ) ⊆ dβ . By Corollary 6.19,
and the above condition that N (bα ) ∩ N (bβ ) ∩ Wsp.cyc ⊆ N (dα − dβ ), we see that
A(bα ) ∩ A(bβ ) ∩ Xcyc ⊆ A(dα − dβ ).
So (A(bβ ) ∩ dα ∩ A(bα ) ∩ Xcyc ) ⊆ dα ∩ A(dα − dβ ). The claim now follows by the above
lemma. The claim implies that S is regular, so we have associated the regular set S to
the global section σ
Conversely, let S ∈ Reg(B) be given. Let {bα } be a family of elements of B which
rectify S and cover Wsp.cyc . Then for each α there exists dα such that
S ∩ A(bα ) = dα ∩ A(bα ) ∩ Xcyc .
Observe that for all α, β:
dα ∩ A(bα ) ∩ A(bβ ) ∩ Xcyc = dβ ∩ A(bα ) ∩ A(bβ ) ∩ Xcyc
as both are S ∩ A(bα ) ∩ A(bβ ). Since A(bα ) ∩ A(bβ ) ∩ Xcyc is a subflow (closed under the
action of t) the above result readily implies that
A(bα ) ∩ A(bβ ) ∩ Xcyc ⊆ A(dα − dβ ).
And by Corollary 6.19, this implies that
N (bα ) ∩ N (bβ ) ∩ Wsp.cyc ⊆ N (dα − dβ ).
But this is precisely what we need to show that the local sections dbα on N (bα ) piece
together to give us a global section σ.
So, to each global section σ we have associated a regular set S and to each regular
set S we have associated a global section σ. A routine check shows that this defines the
desired isomorphism.
COUNTABLE MEETS IN COHERENT SPACESWITH APPLICATIONS TO THE CYCLIC
SPECTRUM
527
Global sections of Γ(Qcyc ) when B need not be countable. We recall the
definition of WC for each countable subset C ⊆ B. As noted above, WC is a spatial
locale for each such countable subset C. Let QC be the restriction of Q, the canonical
sheaf over W , to the subspace WC . The global sections Γ(QC ) can be determined by an
approach strictly similar to the approach in the above theorem. That is, we can define
XC ⊆ X, as the set of all x ∈ X which are cyclic with respect to every c ∈ C. We can
then define a subset of XC to be C-regular by an obvious modification of the definition
of regular (in fact, just replace Xcyc by XC ). The argument used in the proof of 6.24 can
then be used to show that Γ(QC ) is canonically isomorphic to the family of all C-regular
subsets of XC . Then the global sections over the cyclic spectrum, for arbitrary B, can be
described using the following theorem.
6.25. Theorem. Γ(Qcyc ) is the colimit of Γ(Q|WC ) where C varies over the filtered family
of all countable subsets of B.
Proof. We must prove that every global section in Γ(Qcyc ) is the restriction of a global
section in Γ(Q|WC ) for some countable subset C ⊆ B. We must also show that two such
global sections over WC and WD have the same restriction to O(W )cyc if and only if they
have the same restriction to some WE where E ⊆ B is a countable subset with C ∪D ⊆ E.
Clearly, every global section σ ∈ Γ(Qcyc ) is represented by a compatible family
{(dα , bα )} for which σ equals dbα on N (bα ). Since O(W )cyc is Lindelöf, we can assume
that the family is countable and write it as {(bn , dn ) | n ∈ N}. The condition for
the family being compatible is equivalent to a countable set of conditions of the form
N (bα ) ∩ N (bβ ) ⊆ jcyc (N (dbα − dbβ )). But by using Theorem 6.6, this condition holds if and
only if it holds when we restrict to some WC . It readily follows that {(dα , bα )} will be a
compatible family that defines a section in Γ(Qcyc ) if and only if it is compatible enough
to define a section in Γ(Q|WC ) for some countable C ⊆ B. The remaining details are now
straightforward.
7. Examples
7.1. Example of a non-spatial cyclic spectrum. In constructing this example, it
is notationally convenient to introduce, for each n ∈ N, a symbol an and for each f ∈ NN
a symbol hf . We let
G = {an | n ∈ N} ∪ {hf | f ∈ NN }
We let (B, τ ) be the free Boolean flow generated by G.
For each n ∈ N and f ∈ NN we let
U (n, f ) = N (τ f (n) an − an ) ∩ N (τ n hf − hf )
We then claim that:
1. the family {U (n, f )} covers Wsp.cyc ;
528
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
2. the above family has no countable subcover;
3. the cyclic spectrum of this flow is not spatial.
Proof.
/ N such
1. Let I ∈ Wsp.cyc be given. Since I is cyclic, we can clearly define f : N
that τ f (n) an − an ∈ I for all n ∈ N. But, there also must be an n ∈ N for which
τ n hf − hf ∈ I and it follows that I ∈ U (n, f ).
2. Assume there is a countable subcover. Then we can clearly find a sequence (f1 , f2 , . . . fn , . . .)
of functions from N to N such that {U (n, fi ) | n, i ∈ N} covers Wcyc .
/ N such that u(n) > fi (n) whenever i ≤ n. Let v : NN
/N
Now define u : N
be any function for which v(fi ) = i and v(f ) > 0 for all f . Let I be the flow ideal
generated by:
{τ u(n) an − an | n ∈ N} ∪ {τ v(f ) hf − hf | f ∈ NN }
Then I is obviously cyclic, so there exist n, i ∈ N with I ∈ U (n, fi ).
But this implies that τ fi (n) an − an ∈ I and so u(n) < fi (n) which implies that i > n.
On the other hand, τ n hfi − hfi ∈ I which implies that n ≥ v(fi ) so i ≤ n which is
a contradiction.
3. The cyclic spectrum cannot be spatial because, as shown in [Kennison, 2006, Proposition 4.1] this implies that it is a sheaf over the space Wsp.cyc and, by Proposition
6.8, that Wsp.cyc is Lindelöf, which contradicts the above.
7.2. Examples of regular sets.
7.3. Definition. Let (B, τ ) be a Boolean flow. We say that G ⊆ B generates B as a
flow if no proper subflow of B contains G,
7.4. Definition. We say that x ∈ X is periodic if there exists n > 0, such that tn x = x.
We let Per(X) denote the set of all periodic elements of X. We note that Per(X) ⊆ Xcyc .
/ X as the shift map (so that
7.5. Example. Let X = {0, 1}N and define t : X
t(x0 , x1 , x2 , . . .) = (x1 , x2 , . . .)). Let (B, τ ) be the corresponding flow in Boolean algebras.
Then Xcyc = Per(X) is the set of all periodic sequences and every subset of Xcyc is regular.
Proof. Note that g = π0−1 (1) generates B as a flow and that x ∈ X is a periodic sequence
if and only if x is cyclic with respect to g. It readily follows that Xcyc = Per(X).
Let bn = τ n g − g and let S ⊆ Xcyc be any subset. Then we claim that bn rectifies
S. It is readily shown that A(bn ) is the set of all sequences in X which are n-periodic,
which is a finite set. So every subset of A(bn ) is relatively clopen and is clearly of the
form d ∩ A(bn ). It easily follows that bn rectifies any subset S. But the family of all N (bn )
clearly covers Wsp.cyc so S is regular.
COUNTABLE MEETS IN COHERENT SPACESWITH APPLICATIONS TO THE CYCLIC
SPECTRUM
529
7.6. Remark. If (B, τ ) is finitely generated (as a flow) then Xcyc always coincides with
Per(X) and every subset of Xcyc is regular, as the above argument generalizes.
The following proposition is useful in finding regular sets.
7.7. Proposition. Let (X, t) be a flow in Stone spaces and let (B, τ ) be the corresponding
flow in Boolean algebras. Recall the definition of k-Cy(−) from 6.10. Let c ∈ B and the
positive integer k be given. Then:
1. k-Cy(c) is regular;
∩
2. S = Xcyc ∩ n≥0 τ n (c) is regular.
∩
3. S = Xcyc ∩ n≥0 τ n (¬c) is regular.
Proof.
1. Let bn = τ n c − c. It is readily shown that A(bn ) = n-Cy(c). Let S = k-Cy(c). It
follows that S ∩ A(bn ) = (k, n)-Cy(c), where (k, n) = gcd(k, n). A straightforward
argument proves that the set (k, n)-Cy(c) is relatively clopen in n-Cy(c) (as we only
have to restrict the values of ti x for i = 0, 1, . . . , n − 1). A standard argument, using
the compactness of A(bn ) = n-Cy(c), shows that there is a clopen set d of X such
that (k, n)-Cy(c) = d ∩ A(bn ) and, from this, it follows that each bn rectifies S. As
noted in the previous proof, this shows that S is regular.
2. For each k > 0 we claim that bk = c − τ k c rectifies S. Since A(bk ) = k-Cy(c) (by
Lemma 6.14), it suffices to find d ∈ B such that S ∩ k-Cy(c) = d ∩ k-Cy(c) ∩ Xcyc .
But this readily follows for d = c ∩ τ c ∩ . . . ∩ τ k−1 c. Finally {N (bk )} covers Wsp.cyc
since every I ∈ Wsp.cyc must make c cyclic and so contain bk for some k.
3. Note that (2) implies (3) in view of the substitution of ¬c for c.
7.8. Example. Let Σ0 = {σ0 , σ1 , . . . , σn , . . .} be a sequence of “symbols” and give Σ0
the discrete topology. Let Σ = Σ0 ∪ {∞} be its one-point compactification. Let X = ΣN
/ X so that t(x0 , x1 , x2 , . . .) = (x1 , x2 , . . .). Let πn : X
/ Σ denote the
and define t : X
−1
nth projection. For each i ∈ N, let gi ∈ B be defined as π0 (σi ). Let G = {gi }. For this
example, we claim that:
1. G generates B as a flow;
2. Per(X) is a proper subset of Xcyc ;
3. there are regular subsets not of the form b ∩ Xcyc for b ∈ B (so not every global
section over the cyclic spectrum is of the form bb for b ∈ B);
4. not every subset of Xcyc is regular.
Before proving the above claims, we insert a useful definition and some lemmas.
530
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
7.9. Definition. Let (B, τ ) be a Boolean flow and let G ⊆ B generate B as a flow. We
say that p ∈ B is G-prescriptive if there exists g = (g1 , . . . , gm ) ∈ Gm and an m-tuple
k = (k1 , . . . , km ) of positive integers such that
∨
p = p(g, k) =
(τ ki gi − gi )
1≤i≤m
The following lemma shows that, in a sense, a G-prescriptive element of B has the
effect of prescribing a period to an m-tuple of elements of G.
7.10. Lemma. Let p = p(g, k) be a G-prescriptive element of B. Then A(p) is the set of
all x ∈ X which are ki -cyclic with respect to gi for 1 ≤ i ≤ m.
Proof. It is straightforward to show that x is k-cyclic with respect to g if and only if
tn x is never in τ k g − g (for any n ∈ N). The proof then follows.
Recall that ⟨c⟩ is the smallest flow ideal of B which contains c. Also b ∈ ⟨c⟩ if and
only if b misses A(c). Remember that for b, c ∈ B we have that c, A(c) and b are subsets
of X while N (c) and N (b) are subsets of W .
7.11. Lemma. Let b, c ∈ B be given. Then the following are equivalent:
1. N (c) ⊆ N (b);
2. b ∈ ⟨c⟩;
3. A(c) ⊆ A(b).
Proof. (1) ks +3 (2): If N (c) ⊆ N (b), then ⟨c⟩ ∈ N (c) ⊆ N (b) so b ∈ ⟨c⟩. Conversely,
assume b ∈ ⟨c⟩. If I ∈ N (c) then c ∈ I so ⟨c⟩ ⊆ I and b ∈ ⟨c⟩ ⊆ I so I ∈ N (b).
(2) ks +3 (3): Assume b ∈ ⟨c⟩. Then ⟨b⟩ ⊆ ⟨c⟩ which, by the duality between flow ideals of
B and closed subflows of X, implies that A(c) ⊆ A(b). Conversely, assume A(c) ⊆ A(b).
It follows that b misses A(c) so b ∈ ⟨c⟩.
7.12. Corollary. Let S ⊆ Xcyc be given and assume that b rectifies S. If b ∈ ⟨c⟩, then
c rectifies S.
Proof. By the above lemma, A(c) ⊆ A(b) and the result easily follows.
7.13. Proposition. Let (B, τ ) be a countable Boolean flow and let G ⊆ B generate B as
a flow. Let S ⊆ Xcyc be given and let G-Rect(S) be the set of all G-prescriptive elements
that rectify S. Then S is regular if and only if
∪
Wsp.cyc ⊆ {N (p) | p ∈ G-Rect(S)}
COUNTABLE MEETS IN COHERENT SPACESWITH APPLICATIONS TO THE CYCLIC
SPECTRUM
531
Proof. Assume that S is regular. Then
∪
Wsp.cyc ⊆ {N (b) | b ∈ Rect(S)}.
Let I ∈ Wsp.cyc be given. Since I is a cyclic flow ideal, for each g ∈ G, we can choose k(g) >
0 such that τ k(g) g −g ∈ I. Let I0 be the smallest flow ideal containing {τ k(g) g −g | g ∈ G}.
Since G generates B as a flow, it readily follows that I0 is cyclic so there exists b ∈ Rect(S)
such that b ∈ I0 . But for any element b ∈ I0 , there is a finite set F ⊆ G such that b
is in the smallest flow ideal containing τ k(g) g − g for all g ∈ F . Write F = {g1 , . . . , gm },
let g = (g1 , . . . , gm ) and k = (k1 , . . . , km ) where ki = k(gi ). Let p = p(g, k). Then by
the choice of F , we see that b ∈ ⟨p⟩. By the above Lemma, we have p ∈ G-Rect(S) and
p ∈ I0 ⊆ I. Since I is an arbitrary member of Wsp.cyc , it follows that
∪
Wsp.cyc ⊆ {N (p) | p ∈ G-Rect(S)}
The converse is trivial.
Proof of Example 7.8
1. Note that the clopen set πn−1 (σi ) = τ n (gi ). If U ⊆ Σ is a clopen
∧ neighbourhood
of ∞, then F = {i ∈ N | σi ∈ Σ − U } is finite and πn−1 (U ) = i∈F τ n (¬gi ). The
remaining details are now straightforward.
2. Note that x ∈ X is k-cyclic with respect to gi precisely when there exists k > 0
such that xn = σi if and only if xn+k = σi . Let xn = σi if and only if 2i is the
largest power of 2 that divides n. Then x is 2i+1 -cyclic with respect to gi . Since G
generates B, it follows that x ∈ Xcyc . But there is no k such that tk (x) = x as such
a k would have to be a multiple of every power of 2. So x ∈ Xcyc − Per(X).
3. Choose g ∈ G. The subset 2-Cy(g) is regular, in view of 7.7, but is clearly not of
the form b ∩ Xcyc for any clopen subset b ⊆ X (as clopen sets can only restrict xn
for finitely many n). It follows that the global section corresponding to 2-Cy(g) is
not of the form bb for any b ∈ B.
4. Let S be the set of all x ∈ Xcyc such that xn ̸= σm for any even m ∈ N. If S
is regular then, by Proposition 7.7, there are enough G-prescriptive elements that
rectify S. But suppose p is G-prescriptive and that S ∩ A(p) = d ∩ A(p) ∩ Xcyc for
some d ∈ B. Choose gk for an odd k such that gk is not involved in any part of p
or d. Let x ∈ X be the sequence which is constantly σk . Then x ∈ S ∩ A(p) so
x ∈ d ∩ A(p) ∩ Xcyc . But we could just as well have chosen k to be even, in which
case x is still in d ∩ A(p) ∩ Xcyc , but x is not in S, which leads to a contradiction.
532
MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL
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Department of Mathematics and Statistics
McGill University, Montreal, QC, H3A 2K6
Department of Mathematics and Computer Science
Clark University, Worcester, MA 01610
Department of Mathematics and Statistics
Concordia University, Montreal, QC, H4B 1R6
Email: barr@barrs.org, jkennison@clarku.edu, raphael@alcor.concordia.ca
This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/25/19/25-19.{dvi,ps,pdf}
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